# From Static to Dynamic: Adaptive Monte Carlo Search for Mathematical Process Supervision

arXiv: https://arxiv.org/abs/2509.24351 · HTML: https://mlsanity.com/paper/2509.24351/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in Reinforcement learning for reasoning, one of 17 directions in LLM reasoning.
- Builds on 45 of its 48 references that are in the map.
- First posted on arXiv 29 Sep 2025.

## Placed in

- LLM reasoning / [Reinforcement learning for reasoning](https://mlsanity.com/lr/reinforcement-learning-for-reasoning/)
- LLM alignment / RLHF / preference optimisation / [Process reward models for reasoning](https://mlsanity.com/alignment/process-reward-models-for-reasoning/)

## Builds on (most cited first, 30 of 45)

- [Chain-of-Thought Prompting Elicits Reasoning in Large Language Models](https://mlsanity.com/paper/2201.11903/) (2022)
- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Qwen3 Technical Report](https://mlsanity.com/paper/2505.09388/) (2025)
- [Self-Consistency Improves Chain of Thought Reasoning in Language Models](https://mlsanity.com/paper/2203.11171/) (2022)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [Deep reinforcement learning from human preferences](https://mlsanity.com/paper/1706.03741/) (2017)
- [Let's Verify Step by Step](https://mlsanity.com/paper/2305.20050/) (2023)
- [Qwen2.5-Math Technical Report: Toward Mathematical Expert Model via Self-Improvement](https://mlsanity.com/paper/2409.12122/) (2024)
- [STaR: Bootstrapping Reasoning With Reasoning](https://mlsanity.com/paper/2203.14465/) (2022)
- [Math-Shepherd: Verify and Reinforce LLMs Step-by-step without Human Annotations](https://mlsanity.com/paper/2312.08935/) (2023)
- [Solving math word problems with process- and outcome-based feedback](https://mlsanity.com/paper/2211.14275/) (2022)
- [WizardMath: Empowering Mathematical Reasoning for Large Language Models via Reinforced Evol-Instruct](https://mlsanity.com/paper/2308.09583/) (2023)
- [Improve Mathematical Reasoning in Language Models by Automated Process Supervision](https://mlsanity.com/paper/2406.06592/) (2024)
- [Large Language Models are Better Reasoners with Self-Verification](https://mlsanity.com/paper/2212.09561/) (2022)
- [DeepSeek-Prover-V2: Advancing Formal Mathematical Reasoning via Reinforcement Learning for Subgoal Decomposition](https://mlsanity.com/paper/2504.21801/) (2025)
- [Large Language Models for Mathematical Reasoning: Progresses and Challenges](https://mlsanity.com/paper/2402.00157/) (2024)
- [Omni-MATH: A Universal Olympiad Level Mathematic Benchmark For Large Language Models](https://mlsanity.com/paper/2410.07985/) (2024)
- [Rewarding Progress: Scaling Automated Process Verifiers for LLM Reasoning](https://mlsanity.com/paper/2410.08146/) (2024)
- [Language models show human-like content effects on reasoning tasks](https://mlsanity.com/paper/2207.07051/) (2022)
- [MRKL Systems: A modular, neuro-symbolic architecture that combines large language models, external knowledge sources and discrete reasoning](https://mlsanity.com/paper/2205.00445/) (2022)
- [VisualPRM: An Effective Process Reward Model for Multimodal Reasoning](https://mlsanity.com/paper/2503.10291/) (2025)
- [InternLM-Math: Open Math Large Language Models Toward Verifiable Reasoning](https://mlsanity.com/paper/2402.06332/) (2024)
- [Process Reward Models That Think](https://mlsanity.com/paper/2504.16828/) (2025)
- [Let's reward step by step: Step-Level reward model as the Navigators for Reasoning](https://mlsanity.com/paper/2310.10080/) (2023)
- [Paper2Code: Automating Code Generation from Scientific Papers in Machine Learning](https://mlsanity.com/paper/2504.17192/) (2025)
- [MathPile: A Billion-Token-Scale Pretraining Corpus for Math](https://mlsanity.com/paper/2312.17120/) (2023)
- [Improving Large Language Model Fine-tuning for Solving Math Problems](https://mlsanity.com/paper/2310.10047/) (2023)
- [MuggleMath: Assessing the Impact of Query and Response Augmentation on Math Reasoning](https://mlsanity.com/paper/2310.05506/) (2023)
- [JiuZhang3.0: Efficiently Improving Mathematical Reasoning by Training Small Data Synthesis Models](https://mlsanity.com/paper/2405.14365/) (2024)

## Built on by (most cited first, 1 of 1)

- [A Survey of Process Reward Models: From Outcome Signals to Process Supervisions for Large Language Models](https://mlsanity.com/paper/2510.08049/) (2025)

## Abstract (arXiv)

The quality of process data plays a key role in training a Process Reward Model (PRM), which can enhance the complex mathematical reasoning capability of large language models. Existing methods estimate the quality of reasoning steps based on a fixed-budget sampling strategy and navigate a vast search space to perform path expansion during the automated data generation process, resulting in their inefficiency and inflexibility. To address these issues, we propose Adaptive Monte Carlo Search (AMCS), a framework that transforms data generation from fixed, static to adaptive, dynamic search at the level of node value estimation and path expansion. On one hand, AMCS adaptively refines estimation by allocating more samples to uncertain reasoning steps while using fewer samples for those that are easier to estimate. On the other hand, it enhances the path expansion through a Monte Carlo algorithm with a temporally adaptive policy that begins with broad exploration and gradually shifts toward exploiting the most promising directions. With AMCS, we construct a large-scale dataset MathSearch-200K of about 200K process supervision examples for training PRMs. To verify the effectiveness of our method, we conduct extensive experiments on four mathematical reasoning benchmarks. Experimental results show that Qwen2.5-Math-7B-PRM-AMCS achieves up to 76.2% accuracy on MATH500 with GLM-4-9B, outperforming all baseline PRMs. Notably, a 7B model supervised by Qwen2.5-Math-7B-PRM-AMCS surpasses a 72B model with weaker supervision. Moreover, Qwen2.5-Math-7B-PRM-AMCS maintains consistent advantages on out-of-distribution problems, demonstrating strong generalization capability. Our code is available at https://github.com/reml-group/AMCS.